Additiona1ly to primary and secondary rainbow there are unusual rainbows observed over water surfaces. These rainbows are caused by reflected sunlight. Together with the usual rainbows they can form rainbows which at first sight seem to be very sophisticated with up to four partially intersecting rainbows being visible at the same time.

Primary rainbow and reflected-sunlight rainbow at different sun elevations This sketch is also valid for the secondary rainbow although this one has a radius which is about 9� wider.

The formation of the additional primary and secondary rainbows is easy to explain. The water surface has the same effect as a large mirror. The reflection of the sun is as far below the horizon as the sun is above it(the angle of incidence is equal to the angle of reflection). So the point opposite the reflection of the sun (reflected antisolar point) is above the horizon. This point which is above the antisolar point by the double amount of the sun elevation, is the centre of the two reflected-sunlight rainbow circles. So the additional rainbows are vertically displaced in upward direction by the double amount of the sun elevation.

I know about just a handful of photographs of such rainbows from books. In most cases, just a fragment of the reflected-sunlight rainbow is visible at the foot of the primary rainbow. Compared to those, the rainbow fragment in the picture above is rather long. The brightness of this bow, which is about 1/3 of that of the primary bow, is also striking. The picture was taken when the tide sat in on the North Sea coast. Obviously there had been formed lots of rills with a smooth water surface which reflected the sunlight. The photograph makes clear that a reflected-sunlight rainbow can really become a remarkable phenomenon.

The formation of a reflected-sunlight rainbow requires a water surface which is as smooth as possible. Ever when there are only small waves, the sunrays diverge too much to form a visible reflected-sunlight rainbow. That is the reason why no reflected-sunlight rainbow can be seen in the most rainbow photographs over the sea. Good preconditions are found at a lake when there is calm. But the lots of puddles forming after a heavy rainshower might reflect enough light so that it is not absolutely necessary that there is a 1ake or sea near. Perhaps you should try to look for these unusual rainbows. A rainbow can often be predicted about 15 minutes before it appears. Then you should drive to the next lake and wait there for the rainbow with your camera ready. The lake should be in front of the observer when he or she looks into the direction where the rainbow is. There perhaps you can find the additional rainbow fragment at the feet of the usual rainbow. But also with the lake behind your back you can watch the formation of these unusual rainbows. In this case you will probably see the upper part of the reflected-sunlight rainbow.

I also know some old observations which reported of a laterally displaced rainbow appearing additionally to the usual rainbow. As long as there are no photographs available, 1 consider these observations as very doubtful. In this case the reflecting surface should be inclined. Nowadays there might exist such laterally displaced rainbows caused by large ref1ecting house fronts. But I have no explanation for observations made in the l9hh century and earlier. I only could think about very long waves in the sea which might form an inclined plain or of mountain slopes covered with snow and ice.

Rainbow an October 20 1879, at 08.25 hours at Gareloch (Scotland). From his position, the observer Hanrary saw only the first bow (on the left). Other persons nearby saw the whole phenomenon as it is drawn in the picture. Up to now, there are no photographs available of laterally displaced rainbows.From Pernter, "Meteorological Optics", 1st edition, 1906. Pernter took the drawing from "Nature", Vol. 21, Page 56.

Primary rainbow and reflected-sunlight rainbow. After Tait in St. Andrews (Schottland)
on the 11th. September 1874 at 5:40 PM. From Pernter "Meteorologisch Optik",
1. Auflage, 1906. First publication in "Nature" from the 1st. of Oktober 1874.

Exact examination of the reflected-sunlight rainbow

The reflected-sunlight rainbow and the position of the water surface

Figure 1
Water drops forming the rainbow from the observer s �position are all positioned on the surface of a cone with an aperture of 84� (2x42�). Figure 1 shows a section of this cone. The observer s eye is at position "a". The parallel lines are sunrays which are reflected by the water surface. The diagram on the left shows the sunrays at a sun elevation of 10� while the diagram on the right is valid for a sun elevation of 30�. Line "b" points into the direction of the reflected antisolar point. It is supposed that the observer is standing with his or her back towards the sun, looking into the direction where the rainbow is.

You can see from the figure that only those sunrays which had been reflected behind the observer contribute to the upper half of the reflected-sunlight rainbow. The lower half, however, is generated only by sunlight being reflected in front of the observer. So if you are on a beach, it is impossible to see a complete reflected-sunlight rainbow. You will either see the lower half of it when. the sea is in the direction where the rainbow is, or the upper half when the water surface is in the direction where the sun is. Only from a boat, a small island or in other situations when you are completely surrounded by water, it is possible to see a complete reflected-sunlight rainbow.

The length of the light ways and the brightness of the reflected sunlight rainbow

Figure 1 also shows that the light rays forming the upper part of the reflected-sunlight rainbow have to cover a much longer distance than those forming its feet. This has the effect that the reflected-sunlight rainbow becomes fainter in its upper part because raindrops and mist reduce the intensity of the 1ight rays. Photographs of reflected-sunlight rainbows confirm this reduction of light intensity. The feet are always the brightest part of a reflected-sunlight rainbow.

Brightness of a reflected-sunlight rainbow and sun elevation

Figure 2

The brightness of the reflected-sunlight rainbow is in a high grade subject to the sun elevation. Figure 2 indicates the amount of the sunlight reflected from a water surface for vertically and for horizontally polarized light. Additionally it indicates an average which is equal to the naturally, non-polarized sunlight. At low sun elevations the sunlight is almost completely reflected. The higher the sun climbs in the sky, the lesser is the amount of light reflected. The rest of the 1ight is refracted and enters into the water. Just at a sun elevation of 20� only 1/7 of the sunlight is reflected from the surface of the water. That means that the reflected-sunlight rainbow is 7 tines fainter than the normal rainbow and at this sun elevation is only as bright as the secondary rainbow. Furthermore the 1ight becomes fainter by the longer distance it has to run, compared to the 1ight rays generating the primary rainbow as it is described above.

This significant reduction of brightness with increasing sun elevation is the reason for reflected-sunlight rainbows having been observed at low sun elevations only. Under good conditions, however, a reflected-sunlight rainbow should also be visible at sun elevations of 30 degrees and more. The size of the lake reflecting the sunlight also plays a crucial part, and even small waves already cause a significant reduction in brightness of the ref 1ected-sunlight rainbow. These factors have not been taken into consideration here.